chapter 5. underground flow chapter 5. underground flow page 105 5.1 a two-dimensional steady state

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Chapter 5. Underground Flow Page 104

CHAPTER 5. UNDERGROUND FLOW

Contents

5.1 A two-dimensional steady state flow problem 105

5.2 Theory 109

5.2.1 Material data 110

5.2.2 Boundary conditions 111

5.3 An example of transient flow 112

5.3.1 Preprocessing: geometry and boundary conditions 113

5.3.2 Analysis and drivers 115

5.3.3 Material data 116

5.3.4 Results 117

5.4 References 118

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Chapter 5. Underground Flow Page 105

5.1 A two-dimensional steady state flow problem

Dupuit flow towards a trench is a common problem in civil engineering. Let’s consider

the example shown in Fig. 5.1. It corresponds to a steady state flow with known

upstream and downstream head.

Fig. 5.1 Flow towards a trench, geometry

ZSOIL DATA: Ex_5_1_Dupuit_flow.inp

Let’s open the input file and examine the data (Fig. 5.2). Under Control/Analysis &

Drivers, we see that the problem is defined as Plane (strain) flow, with a Time

dependent/steady state driver. A single time step is sufficient in this case since flow is

steady, but iterations (within the step) will be needed to find the free surface.

Fig.5.2 Driver for flow analysis

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Chapter 5. Underground Flow Page 106

Under Assembly/Preprocessing we discover the mesh and boundary conditions. There

are 3 types of boundary conditions:

- “no flow” on the upper and lower edge; this is a default boundary condition,

which does not need any input.

- Imposed pressure: select FE model/Boundary Condition/Pressure

BC/Update Parameters , click on the left boundary; if parameter screen does

not pop up, use scale, change sign, then diagram show reverse, then repeat:

update parameters and click again on pressure diagram, values can now be

updated.

- The third type of boundary condition is a seepage condition, it is implemented as

a seepage element: go back to FE model/Seepage/Direction/Show a set of

black line segments will point outwards of the seepage boundary (Fig. 5.3). This

type of boundary condition is used whenever a seepage out- or inflow can be

expected. The boundary condition will impose a flow proportional to the difference

of pressure between in- and outside the boundary, if the medium is saturated

inside, and a “no flow” condition if the medium is not saturated inside.

Exit preprocessor now without saving.

Seepage boundary condition

Pressure boundary condition

no flow

Flow % pressure difference

Seepage boundary condition

Pressure boundary condition

no flow

Flow % pressure difference

Seepage boundary condition

Pressure boundary condition

no flow

Flow % pressure difference

Fig. 5.3 Flow boundary conditions

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Chapter 5. Underground Flow Page 107

Under Assembly/Materials we observe the presence of 2 materials:

Continuum/Elastic and Seepage/Seepage. The first one is there only because we

solve the flow problem on the whole (solid) domain using partial saturation, default

values of E , νννν and γγγγ can be left as is, they have no influence.

Flow under steady state requires the permeability KDarcy (which can be oriented is space

with angle ββββ) and γγγγF, the fluid unit weight to be specified. In addition, due to possible

partial saturation, (1/αααα) the thickness of the transition from full to residual saturation,

and Sr the residual saturation must be given. Other data, default or not, can be ignored.

Fig. 5.4 Material data for Continuum

Fig. 5.5 Material data for Seepage

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Chapter 5. Underground Flow Page 108

Seepage requires specification of Kv, a penalty factor, multiplier of K [m 3/(Ns)], an

internally optimized permeability factor which will regulate the outflow through the

seepage surface. Leave default value = 1, and see theory manual for details.

Analysis/Run Analysis can be performed in 1 step, but a few iterations are needed to

find the free surface.

Results/Postprocessing/Graph Option/Maps. After running the problem, the plot

of pressure maps indicates the position of the free surface, and settings can be changed

to improve the visibility. Settings/Graph Contents, with imposed scale (Min = 0,

Max = 1), indicates a free surface which corresponds to what is expected (Fig. 5.6).

Remarks:

- The free surface does not change too significantly in this problem, when Kv is

changed, but the outflow does. When Kv is increased permeability and therefore

outflow increases.

- The free surface corresponds to zero pressure, pressures above the free surface

are positive, but their contribution to total stress in the medium is multiplied by

the saturation, so that the effective water pressure above the free surface is in

fact zero: 'totij ij ijSpσ σ δ= + and 0Sp ≅ above the free surface.

Fig. 5.6 Color maps of pore pressure

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Chapter 5. Underground Flow Page 109

5.2 Theory

Underground Darcy flow is governed by a diffusion equation, expressed here in terms of

pressure as the nodal unknown. The equations for partially saturated Darcy flow are

given in Table 5.1.

The approach adopted in ZSOIL considers that the flow domain coincides with the soil

domain, always, saturation will ultimately define the limits of flow. The formulation

accounts for partial saturation and the free surface corresponds to the p=0 line in Fig

5.6.

In table 5.1, the continuity equation expresses that a local source induces a divergent

flow divergence and a local time-dependent pressure variation. The flow conditions can

be transient, in this case all terms of the continuity equation are active, or steady

state, in which case there is no pressure change in time and the corresponding term in

the equation, cp& , can be ignored.

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Table 5.1 Equations of partially saturated underground flow

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If the flow is confined, the medium fully saturated, and the flow condition steady, then

the problem is linear; it can be solved in a single step without iterations. But most of the

time the exact position of the free surface is unknown a priori, it is part of the problem to

be solved, the problem becomes therefore nonlinear and iterations are needed even for

the steady state flow, a fortiori within each time-step in the transient case.

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Chapter 5. Underground Flow Page 110

5.2.1 Material data

Material data needed include Darcy’s coefficient K, and γγγγF, the fluid’s specific weight, in

the basic version; all other parameters are selected by default. In the advanced

version the fluid compressibility ββββF and two new coefficients are introduced, (1/αααα) [m], a

measure of the thickness of the transition from full saturation to residual saturation, and

Sr which is the second parameter.

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Table 5.2 Material data for flow problems

3

[ / ] : ' ,

[ / ] :

( . . ) :

[ ]:

m s Darcy s permeability coefficient often a scalar

N m fluid specific weight

e g free surface and

m thickness of transition from ful

ij

F

F

Steady state saturated flow

K

Steady state partially satu

K

γ

K,γ

(

rated flow

1/α)

2

[ ]

: constant

( . . ) :

[ / ] : mod

[ ] : ( . / . ),

/(1 ) :

l to residual saturation

seepage multiplier

e g free surface

N m fluid bulk ulus

void ratio vol of voids vol of solid from which

n e e po= +

r

v

F r v

F

o

Transient partially saturated flow

S

K

K,γ ,α, S ,K

β

e

,

(1 ) / . 2

/ . ; / . w s

w s

rosity and

nS n mass unit vol of phase medium

mass unit vol of water mass unit vol of solid

ρ ρ ρ ρ ρ

= + − = = =

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Material data αααα and Sr depend strongly on the granulometric structure of the medium,

values taken from literature are given in Table 5.3.

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Table 5.3 Flow parameters from [Yang & al. 2004]

Type of soil αααα Sr

Gravely Sand 100 0

Medium Sand 10 0

Fine Sand 8 0

Clayey Sand 1-1.7 0.23-0.09

[Yang & al.